Note for the Fermat Equation

The Fermat's Last Theorem was first stated as a theorem by Pierre de Fermat around 1637 in the margin of a copy of Arithmetica. Fermat added that he had a proof that was too large to fit in the margin. He claimed to have discovered a proof for the equation $a^{n} + b^{n} = c^{n}$ having no solutions in positive integers for $n$ greater than $2$. However, he didn't provide the details of his proof. This theorem remained unproven for centuries until Andrew Wiles published a proof in 1994. Wiles proof is very far for being closed to the Fermat's claimed theorem due to its long extension, complexity and tools that were only available during the 20th century. This work could be closer to the Fermat's claimed proof.